A model for the nonautonomous Hopf bifurcation

TL;DR

This paper rigorously analyzes the nonautonomous Hopf bifurcation, demonstrating the emergence of a generalized torus and bifurcation patterns in both deterministic and random forcing scenarios.

Contribution

It provides a thorough, rigorous analysis of the bifurcation pattern in nonautonomous systems, including the existence of a generalized torus in both deterministic and stochastic settings.

Findings

Existence of a generalized torus after bifurcation

Bifurcation pattern characterized in deterministic models

Extension of analysis to randomly forced systems

Abstract

Inspired by an example of Grebogi et al [1], we study a class of model systems which exhibit the full two-step scenario for the nonautonomous Hopf bifurcation, as proposed by Arnold [2]. The specific structure of these models allows a rigorous and thorough analysis of the bifurcation pattern. In particular, we show the existence of an invariant 'generalised torus' splitting off a previously stable central manifold after the second bifurcation point. The scenario is described in two different settings. First, we consider deterministically forced models, which can be treated as continuous skew product systems on a compact product space. Secondly, we treat randomly forced systems, which lead to skew products over a measure-preserving base transformation. In the random case, a semiuniform ergodic theorem for random dynamical systems is required, to make up for the lack of compactness.